There is no doubt that one of the main goals of Analysis is to study the asymptotic behavior of functions. This is known as “Asymptotic Analysis.” It has applications in many branches of science, including physics, and engineering. It also has applications in both pure mathematics (e.g., number theory) and applied mathematics (e.g., differential equations). The main results of asymptotic analysis are not usually treated in classical analysis books; for this reason, it is very useful to have an account of those results in this friendly volume. The book under review is a very good reference on this material, giving a detailed collection of various asymptotic results, with a special focus on special functions.

This book centers on integrals and the solution of differential equations. Chapter 1 gives fundamental tools in asymptotic analysis of real- and complex-valued functions. It contains important and useful results, including some on asymptotic solutions to transcendental equations. Chapter 2 introduces the definitions and main properties of several frequently-used special functions. Chapters 3, 4, 8 and 9 focus on approximation of integrals and sums. Various real and complex tools are presented. Some good examples are given from combinatorics and some other branches of mathematics. Chapters 5, 7, 10, 11, 12 and 13 focus on the studying approximate behaviors of the solution of differential equations.

The book is a classic, and it seems to be essentially a research text, but it has the structure to be also used as a textbook. Indeed, each section includes good and challenging exercises, some of which are the key and starting point for further research. At the end of each chapter, we find a brief historical note and additional references.

To have a good understanding of the content of book, one needs some knowledge of undergraduate analysis, including foundation of sequences and series and the theory of complex valued functions. Some knowledge of differential equations is also required. Thus, this is a book for graduate students; however, it can be used as a complementary source for undergraduates, in which they will be able to find some delicate examples for their courses in analysis and differential equations.

This impressive book contains more than what appears in its table of contents; the reader will find much that is very nice and useful inside it. I recommend it strongly for students and professors of mathematics, physics and engineering who are concerned with careful analysis of asymptotics and special functions. For example, students in analytic number theory, which uses various properties of special functions over the complex field, will find much that sheds light on various formulas and methods in analytic number theory.

Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His field of interest is Elementary, Analytic and Probabilistic Number Theory.

## Comments

## Warren Johnson

"Friendly" is not one of the first 500 adjectives I would choose to describe Olver's book, but it's still not only the best book on asymptotics after 37 years in print, but also one of the better books on special functions.